x^2=x+54

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Solution for x^2=x+54 equation:


Simplifying
x2 = x + 54

Reorder the terms:
x2 = 54 + x

Solving
x2 = 54 + x

Solving for variable 'x'.

Reorder the terms:
-54 + -1x + x2 = 54 + x + -54 + -1x

Reorder the terms:
-54 + -1x + x2 = 54 + -54 + x + -1x

Combine like terms: 54 + -54 = 0
-54 + -1x + x2 = 0 + x + -1x
-54 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-54 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -1x + 54 + x2 = 0 + 54

Reorder the terms:
-54 + 54 + -1x + x2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -1x + x2 = 0 + 54
-1x + x2 = 0 + 54

Combine like terms: 0 + 54 = 54
-1x + x2 = 54

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 54 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 54 + 0.25

Combine like terms: 54 + 0.25 = 54.25
0.25 + -1x + x2 = 54.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 54.25

Calculate the square root of the right side: 7.365459931

Break this problem into two subproblems by setting 
(x + -0.5) equal to 7.365459931 and -7.365459931.

Subproblem 1

x + -0.5 = 7.365459931 Simplifying x + -0.5 = 7.365459931 Reorder the terms: -0.5 + x = 7.365459931 Solving -0.5 + x = 7.365459931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 7.365459931 + 0.5 x = 7.365459931 + 0.5 Combine like terms: 7.365459931 + 0.5 = 7.865459931 x = 7.865459931 Simplifying x = 7.865459931

Subproblem 2

x + -0.5 = -7.365459931 Simplifying x + -0.5 = -7.365459931 Reorder the terms: -0.5 + x = -7.365459931 Solving -0.5 + x = -7.365459931 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -7.365459931 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -7.365459931 + 0.5 x = -7.365459931 + 0.5 Combine like terms: -7.365459931 + 0.5 = -6.865459931 x = -6.865459931 Simplifying x = -6.865459931

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.865459931, -6.865459931}

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